Livestrong day's third and fourth stops are equally energetic, with a turnout of hundreds clad in yellow at the University of Denver and a swishy dinner at Wolfgang Puck's Cut in Las Vegas. ("I have the same motto as Lance: to live strong, except I also say, 'Live strong, and eat and drink well,' " declared Puck, raising a glass of champagne.) By the end of the day everyone on the plane is completely exhausted. We're all crashed out in our seats, but Armstrong's still roving around, ready to talk. At 36 he has the energy of a teenager, like the 15-year-old professional triathlete he once was. He even thinks about reentering triathlons these days but feels as if he's already been there, done that. "Plus, if I did enter, I would have to win," he says, with another of his shit-eating grins.

We have a brief talk about politics, but Armstrong is so touchy – and canny – that he asks me to turn off the tape recorder even to talk about the war in Iraq ("We gotta git Osama, the sumbitch," is as much as he'll say on the record). He refuses to disclose his party affiliation, though he has said he's pro–gun control, pro-choice, against the Iraq war, and all for maintaining the line between church and state. He's obviously deeply immersed in learning the way the game is played, although it's a recent immersion. When I bring up Mike Huckabee, he almost leaps out of his seat. "You can't like him!" he shrieks. "Paul Begala said that Huckabee is so conservative he doesn't believe in photosynthesis, evolution, and gravity. I mean, he's charming and smart and quick on his feet, but jeez!"

To take Armstrong seriously as a candidate, you'd have to think that he put away a lot of his famously hardheaded ways. He wants to convince you of that, but in a sense he is a little bit isolated because his life experience has been so different from his peers'. He retired at 33, the time when most guys in this country are hitting their prime. "Cycling was like therapy for me, being alone with my thoughts and the elements," he says. "I had 15 great years of those therapy sessions. I try to keep life simple, to get things out if I need to get them out, but sometimes I just put something in a box in the corner and then it sits there, going, 'Fuck you.' If I can't figure it out, I'll go talk to this gal in Austin. She's a therapist, okay, but she's more like a drill sergeant."

Armstrong is an icon who gets to meet a lot of other icons, but he's not sure if he should treat them like icons or friends, and a lot of them take the world far more seriously than he does. "When I finished the Tour, Bono's advice to me in starting my foundation was that I should take a year and go talk to whoever I wanted to talk to – Bill Gates, or Rupert Murdoch, or whoever," says Armstrong, who modeled much of his funding drive for cancer on Bono's support for Africa. "Now, I've done some of that, but that takes a lot of time, and I can't just put everything on hold so I can have coffees with people all around the world. And I'll tell ya, there are times where I do not talk about cancer – the foundation, my diagnosis, my recovery, nothing. Bono is always on. He's always on."

The fact is that in addition to success, achievement, and competition, Armstrong just really likes to have fun. He was divorced from his wife Kristin, a pretty blond author, in 2003, after five years of marriage; he called off his engagement to Sheryl Crow in 2006, after they'd dated for almost two years. These days he's unapologetic about his reputation in the tabloids as a pussy hound, linked to designer and socialite Tory Burch, Ashley Olsen (whom he denies dating), and, until recently, Kate Hudson. "I'm a 36-year-old single guy who is completely open and honest with every woman in my life," says Armstrong. "As long as you're honest, and no one's getting played or let down, then you're being fair."

Armstrong doesn't go on a lot of real dates. "I think it's hard for me, or for anyone in my position, to call someone up and say, 'Hey, can I take you on a date?' " he says. "It's never like that. I meet people casually, while hanging out with friends, so there's less pressure." Sex is something he enjoys, because he didn't get much of it on the Tour. "I had sex if I had the energy. I wasn't one of those guys who believed in the myths about the guy losing his chi," says Armstrong. "But the fact is that if you are riding your bike five, six, seven hours a day, you are not a sex champion. You're just not. You have fatigue, low testosterone, and a lower libido." He grins. "But you know, I never got any complaints."

Armstrong retreats from questions about Hudson, whom he dated for a few months. "I made a mistake to let the public into my relationship with Sheryl, even though it didn't feel like a mistake at the time," he says. "It put a lot of pressure on the relationship, and I will never do it again." Some public outings can't be avoided, though. "When Tory had a show, I would go to the show; when Sheryl had a concert, I'd go to the concert; and if – uh – a new lady has a premiere, I might go to the premiere. I'm there as a supportive partner, and that's my role. But sitting down and doing a bunch of photos, like I did with Sheryl, is just wrong, and I won't do that again."

It is hard to miss that most of these women are somewhat similar: down-to-earth blond lasses with Cheshire cat grins and small waistlines. Does he have a type?

"Everyone says that!" says Armstrong, moodily. "Everyone says I have a type! Of course I have a type. But doesn't everyone have a type? So shut up!"

Um, isn't that type a little like your mom?

"Now, hold on a minute there," he says, laughing. "That's just gross."